Problem: The arithmetic sequence $(a_i)$ is defined by the formula: $a_1 = 15$ $a_i = a_{i-1} - 4$ What is the sum of the first 12 terms in the series?
The sum of an arithmetic series is the number of terms in the series times the average of the first and last terms. First, let's find the explicit formula for the terms of the arithmetic series. We can see that the first term is $15$ and the common difference is $-4$ Thus, the explicit formula for this sequence is $a_i = 15 - 4(i - 1)$ To find the sum of the first 12 terms, we'll need the first and twelfth terms of the series. The first term is $15$ and the twelfth term is equal to $a_{12} = 15 - 4 (12 - 1) = -29$ Therefore, the sum of the first 12 terms is $ n\left(\dfrac{a_1 + a_{12}}{2}\right) = 12 \left(\dfrac{15 - 29}{2}\right) = -84$.